Compact sets

Originally Posted by Panconleche
Hello,
I am taking the first part of Topology and I am having a little trouble understanding the concept of a Compact Set. The biggest problem to me is the definition of Compact, open cover, and sub cover. I have read the material and try to make sense of it, however, I am still a little lost with the material. If you could shine some light my way I would appreciate it.

Thank you
A subset H of a Euclidean space X is said to be compact (in X) if every open covering of H has a finite subcovering.

Here are a few examples.

Consider the interval $[1,2]$ can and the cover $\{ 1-\frac{1}{n},2+\frac{1}{n}\}_{n \in \mathbb{N}}$

This covers [1,2] for all values of n but it also covers for any finite value of n.

Now consider (1,2) and the cover above this covers it, but it must be true for every covering.

Now consider $(1+\frac{1}{n},2-\frac{1}{n})_{n \in \mathbb{N}}$
at "infinity" it covers (1,2) but doesn't for any finite value of n.
so (1,2) is not compact

I hope this helps.

Brett